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no. 40-49 

cop. 2 

Digitized by the Internet Archive 
in 2013 


Technical Report No. 48 


Dr. Y. T. Lo 

Contract AF33 (616) -6079 
Project No. 9-(13-6278) Task 40572 

Sponsored by. 


Electrical Engineering Research Laboratory 

Engineering Experiment Station 

University of Illinois 

Urbana, Illinois 




It is a pleasure to acknowledge the comments on the manuscr]p T f 
this report gnen by Professors G, A Deschamps and P E Mayes 



1„ Introduction 1 

2. Results and Discussions 4 

3. Simplified Analysis 18 
Conclusion 28 
Reference 29 





1. Cross-section of a zoned mirror 2 

2„ Image pattern on the focal plane for a = 6 

3„ Image pattern on the focal plane for a = 5° 7 

4. Image pattern on the focal plane for a = 10° 8 

5„ Image pattern on the focal plane for a = 15 9 

6. Image paMprn on the focal plane for a = 20° 10 

7 a) maximum field intensity of the image vs the angle of 

incidence a, b) variation of directive gain vs scan angle 

for the designed frequency f and 1.0625 f 12 

o o 

8„ Relation between the position of off-axis feed and beam 

deflection angle a at 1„0625 f 13 


9. Ratio of the two first secondary maxima vs scan angle a 

for the designed frequency f and 1.0625 f 14 

10. Magni + ude of secondary maximum vs its position in the 

focal plane for the designed frequency f and 1.0625 f 16 

o o 

11, Geometry cf a typical zone of the mirror 19 


Briefly, a zoned mirror consists of sections of a set of confocal 

parabolas with focus at c and axis VC, defined by P=2(f-nX./2)/(l+cos4 J ), 

n = 0, 1, 2, ... where f is the focal length of the parabola with the 

largest focal length as shown in Figure 1 „ This family of parabolas 

intersect the circle 2 with center also at C and radius f at V , P , P„.c « 

1 1 2 

If a set of parallel planes perpendicular to the axis VC are drawn through 

the vertices N , N„, N„, „ „ „ of the parabolas, they will intersect the 
1' 2 3' 

circle at Q , Q , Q , „ . , The sections of the parabolas cut out by the 
lines parallel to VC and passing through Q , Q , Q , » . . form a zoned 

mirror as shown by the solid line in Figure 1„ 

In a previous report the properties of such a coma-corrected zoned 

mirror are investigated by diffraction theory. The image pattern for various 
incident angles of a plane electromagnetic wave and also the radiation 
characteristics of the system are evaluated numerically by means of simple 
fundamental functions and Fresnel integrals, It is concluded that for a 
mirror with small F-number and nearly uniform illumination the zoned mirror 
shows great effectiveness in coma correction. Since for such a two- 
dimensional cylindric mirror there is no spherical aberration and astigmatism, 
the only important defect of this system will be the chromatic aberration. 
Unfortunately, such a system is inherently very frequency-sensitive; moreover, 
to this author s knowledge there is yet no method available for chromatic 
correction of this system. However, in contrast to many optical systems 
a microwave device even with a limited bandwidth usually finds very wide 
applications In this investigation the image pattern deterioration due to 
chromatic aberration is studied. The purpose of this short report is to 

/ B 2^ 2 - 

Figure i. Cross-section of a zoned mirror 

supplement some of the information on this point which is not found in 
the previous report. 


Diffraction theory was used in the previous *ork c 1 nee coma aberrati >n has 

a dominant effect on the minor lobes where the usual hirchhoff approximation 

is generally susceptible to larger error However the results sho* that 

the contribution to trie image field due to the coupling current? among 

zones is negligible in comparison with that due to the geometric optics 

current induced by an incident plane wave, Moreover. t he edge diffract. on 

effect is also of higher order as compared to the latter (except at nulls 

and small minor lobes), However, with considerable chromatic aoerration the 

lower order contribution becomes appreciable (the nulls oi 'he pattern will 

be filled up and the minor lobe level will be raised) therefore, tne 

above mentioned high order effects become less significant when the mirror 

is operated at a frequency other than that of proper design In such a 

case the previously established theory'can be greatly simplified, since 

it requires no near-field solution which is only essential in the evaluation 

of coupling effed between zones, Such a simplified theory will be 

established and discussed in the next section However., this theory is 

not used here to evaluate the deterioration of the patterns =ince a 

program for the ILLIAC (the electronic computer at the University of 

UlinoJ ) based upon the more rigorous approach ha? already been established 

th< previous investigation. With only a minor modification this 

idapted to the evaluation of the performance of a zoned 

dimensions at frequences equal to 1 0625 I 


i ! 

(a) F-number = f/D = 0.556, 

(b) the focal length = f 10\ where X. is the wavelength corresponding 
to the design frequency f ; 

(c) total number of zones = 11. 

For the purpose of comparison, the performance of an equivalent smooth 
parabola (one with same focal length and aperture) is also computed. As 
expected the latter by comparison is a wide-band device since the only effect 
of changing frequency is to vary the aperture dimension in wavelength accord- 

Figures 2 and 6 show the image pattern in the focal plane for an incoming 

o o o o o 
plane wave with incident angle equal to , 5 , 10 , 15 , and 20 with respect 


to the axis of the system. By comparing these patterns with those at the 

design frequency f , it is seen that at a frequency deviated by 6.25 7c from 

f the image patterns are changed considerably. First of all the pair 

of minor lobes adjacent to the major beam are no longer distinct; instead they 

are merged with the major lobe. This is evidently a result of overwhelming 

chromatic aberration. However, it is interesting to see that other minor 

lobes seem to be comparatively less affected by this aberration and also that 

the symmetry of the image pattern is maintained up to an incident angle 

equal to 10° „ For the smooth parabola, the image patterns remain very much 

the same as those at the design frequency f Q , except that the beamwidths are 

somewhat smaller and the gain is slightly higher up to a scan angle o. = 10 f 

this being a consequence of an increment in aperture as referred to 

wavelength. However, this is no longer true for large scan angle since a 


*> ^ ro cvj 

(AJLISN31NI 01313 3AllV13d) z 3 






(A1ISN31NI Cl~l3ld 3AI1V1V13U) z 3 

(A1ISN31NI 01313 3AI1V13U) Z 3 








in <fr ro cj 

(A1ISN31NI 01313 3/VI±V13H) Z 3 


sr ro cj 

(A1ISN31NI Q13IJ 3AllVH3d) z 3 


system of wider aperture has larger aberrations. These results can be seen 

in Figure 6 which also shows the maximum field intensity (equivalent to the 

gain except for a proportionality constant) and the normalized gain (referred 

to the gain at a = 0) against the scan angle a for both mirrors at frequencies 

f and 1.0625 f . 
o o 

It is seen from Figure 7(a) that the loss in gain of the zoned mirror 

due to 6.25% increment in frequency is from 1 db at a = to about 2 db at 

a = 20 . Figure 7(b) also shows that the gain at this frequency drops at 

a faster rate than that at f as the scan angle a increases (somewhat close 


to that of the smooth parabola) . 

Figure 8 shows the relation between the position of the off-axis feed 

and the beam deflection angle a at 1.0625f . If these curves are compared 


with those at f , one finds practically no difference. This fact indicates 

that the lateral chromatic aberration of the so-called "diffraction focus" 

is not severe at all, at least in the range considered. As to the longitudinal 

chromatic aberration no computation has been done. But as will be seen , later, 


a very simple result can be obtained which states that as expressed in*" 

percentage of the original focal length the longitudinal chromatic aberration 

is equal to the percent change in wavelength as referred to \> . 

Since the first pair of minor lobes are not distinct, in other words, 

the coma aberration is overshadowed by a strong chromatic aberration at 

this frequency, it becomes meaningless to evaluate the coma in terms of 

their ratio as has been done previously. However, for the smooth parabola 

this ratio is shown in Figure 9 at both frequencies together with that for 

the zone mirror at f . Since a larger ratio indicates a larger coma aberration 


this result is in agreement with the fact that a wider aperture (in wavelength) 




UJ f- 

Q X 


'■0625 ft 



10 15 20 







10 15 20 



I 7. a) maximum field intensity of the image vs the angle of incidence Q, 

b) variation of directive gain vs scan angle for the designed frequency 

f and 1.0625 f 
o o 





ro cvj 

y/ k lN3IAI30V~ldSIQ Q333 









o o 


o m 
•h eg 

+- 1 to 

■H O 

O ,-i 


0) a! 




ce -a 


suffers more from aberrations. 

Figure 10 shows the field intensity levels of the first pair of the 

minor lobes (adjacent to the major beam) vs, their positions in the focal 

plane of the smooth parabola for an incident wave at frequencies f and 


1.0625f . Also shown in this figure are those of the zoned mirror at f 
o o 

only. It may be noted that the side lobe level of an image pattern is 

not exactly the same thing as that of the corresponding radiation pattern. 

The purpose of preparing Figure 10 is in an effort to relate these two 


By reciprocity if a feed is at y/^ in the focal plane minor lobes of 

corresponding intensity as shown in Figure 10 will be produced somewhere in 

space. At the same time the major lobe field intensity corresponding to this 

feed displacement can be found by first using Figure 8 to obtain the scan 

angle # ( or the beam deflection angle in the terminology of transmitting) 

for this value of y/^ , then referring Figure 7(a) for the intensity. The 

ratio of these two intensities is the side lobe level of the radiation 

pattern for the feed at y/^ . Incidentally Figure 10 also shows slightly 


higher side lobe levels for the parabola operating at higher frequency, 
especially at large scan angle. 

In FiguVes 2 to 6 there are also shown the image patterns of the zoned 
mirror at a frequency of 1 . 125f . Except the case of normal incidence all 
images are so blurred" that the major and minor lobes are no longer 
distinguishable A poor image in this case may be easily explained. At 
this frequency the field contribution of the outer six zones (although 
they are narrower in width) will have a component opposite to that due to 
the central zone for the normal incidence case. On the other hand at a 


frequency equal to l,0625f , no such negative contribution arises except the 
outer two narrow zones. If one uses this as a criterion to determine the 
bandwidth such that no zone contributes a field at focus in opposite sense 
then the maximum total bandwidth as referred to the frequency f is given 

by — = .- — where N = total number of zones. For the present mirror 

f N-l ^ 


A f/f = O.lOo No computation for exactly this frequency has been done 
since this criterion does not necessarily imply only a slight deterioration 
of the pattern and a small minor lobe which are of primary interest for 
.scanning purpose. It seems to be safe to infer that for the latter application 
the bandwidth will be less than the value indicated above. 

In conclusion, it is seen that from the point of view of the image 
sharpness, the gain, and side lobe level, the zoned mirror at 1.0625f or 
higher is worse than a corresponding smooth parabola in scanning ability. 
A similar conclusion may also be reached on the lower frequency side of f „ 



Since for the zoned mirror at a frequency different from f , the design 
frequency, the chromatic aberration becomes so overwhelming, a lower order 
approximation will provide a fairly accurate solution. In this section 
such a simplified theory will be given. 

In Figure 11, a typical zone, (except the central one) is shown *ith a 
plane wave incident at an angle a with the axis ox. Let the intersection of 
the zone and the circle be Q (x ' , y' ), a typical point on the strip be 
QCx', y') , the angle of the strip with respect to the y-axis be T and 

0Q o = s > 0Q= s ^ 

Then the current induced on the strip based upon the infinite plane 
solution in the previous report . 

ICQ) = I cos (« + x) e jk S Sin (a + T) Cl) 

where *1 1 s the intrinsic wave impedance of free space , The f leld intensi ty 

at observation point P (x,y) is given by 

E(p) = -i--cos (a + T) f 2 H (2 Hr) e jkS Sin {a + T) ds (2) 
2J 7 ! J o 

S l 

where s and s are distances of the edges of the strip function origin 
1 ^ 

0, and 

f f ' N 2 / / N 2 

r = | (x - x) + (y - y) 


In fact this is generally true in optical systems c In systems with 
large aberrations even geometrical optics gives a good description of 
the image deterioration, See E Wolf, Rep Prog in Phvsics (London 
Physical Society) 14, (1954), 95 Also~M BorrTand E„~Wolf , Principles 
of Opt ics. Chapter IX, 1959, Pergamon Press 



Figure 11. Geometry of a typical zone of the mirror 



x ! = x' + x , y' = y' + y 

o o 

2 2 V2 
s" = (x- 2 + y" 2 ) 

r = P Q , 

o o ' 


5 = the angle between P Q and the axis x 0. 


oo 2 

r = r - 2r S" sin T cos 6 + 2r S" cos T sin 6 + S' (3) 
o o o 

= r 2 + S*' 2 + 2r S" sin (6 - t) 

o o 

For the observation point P in the neighborhood of focus F such that 

r ^ f >> fS - S)/2, then 
o 2 1 

v m r [1 + S" sin (6 - T)/r ] (4) 

o o 

Since we are primarily interested in large aperture which implies that the 

focal length f is considerably larger than wavelength X^ the Hankel function 

in the integral can be replaced by its asymptotic expression 

r~\ m — -, . „,. I „ ■ jkr [1 + s" sin (5 - r)/r 

H^) ( k r) « PL e -^ + J */4 * /_£L- e J o L 
o f kr WTTkr 



o~ /-a T % "Jk[r - S sin (a + T )] f S 2 ., „ 

w v c os (a + t ) o o J -Jkqs , „ 

E(p) = ; 07r . , ~ e J „ e J M ds 
^ikr -^ s 

cos fa + Tl -J fc t'o " S o Sin (a + T) ^ ^"^^ - ^^/J^ 
t| /27Tjkr e 


where q = sin f6 - t) - sin ( a + t). 


By assuming that s'' = - s" which is nearly true for most strips (or by 

redefining the point Q as the center of the strip, not the interaction with 

o > 

the coma circle) then 

., , /Tcos (a + x) sln < k 1 S 2> -JK[r o - S o sin(a + x)] 

E<P) = n^ST — e 

This is equivalent to the field due to an inhomogeneous line source located 

at Q with a current 

o "H kq 

since q is a function of 6, depending on the coordinates of observation point P. 

Let, t 5 , q , s , r , and s" (half width of the strip) be those 
' n' n' n' on' on' n 

values for strip number n. The total field at P(x^y) is 

N /2 cos (a + t ) sin(kq s" ) -jk[ r - s sin (a + t )] 
*, \ *, , \ ^ n nn on on n J 

E(p) = E (p) + 2 — e 

n=-N T] /j77kr on n (g) 


where E (p) is the field at P due to the central zone which has been obtained 
previously. To conform with our present notations, it is rewritten as follows 

■* , x /jf~~ f 1 cos(4j/2 + a) -jk[r-f tan iJj/2 (tan i|i/2 cos a + 2 sin a)] 

^1 (P) =t\ \ 2 , /Q e 

v J _ cos 4V2 


= f sec ty/2 + P - 2fp sec \\>/2 cos (9 - i|0 

(P,Q) is the polar coordinate of P(x, y) with respect to the focus F 
and axis FO. 


20 is the angular aperture of the central zone (parabola) referred to F. 
For P in the neighborhood of F, f > > P, then 

r *» f sec 4V2 - p cos (9-4) 

Let z = tan 4V 2, and the coordinates of P with respect to F be (Ax, y) ; then 


dz = dLJJ/2 cos i|j/2 

cos 4j = (1 - z ) / (1 + z 2 ) 

2 2 

sin (^ = 2z / (1 + z ) 

2 2 

cos 4*/2 = 1/(1 + z ) 

2 2 2 

sin i^/ 2 = z /(I -*- z ) 

Substituting these quantities into (9), one obtains 


v ( \ n h f cos a - z sin a. -jk(p(z) . . . 

E 1 (p)=2 ^ - / ^ — eJr dz (10) 

1 J -Z V 1 4- z 


z 2 = tan Q (10) 

<p(z) = i x-2(f sin a. + y ) z + [f(2-cos a) + A x]z -2f sin a z + f(l-cos a.) z 4 ] 

„ 2.-1 (11) 

X (1 + z ) 


z 2 = I (X/f - X 2 /4f 2 ) (12) 

Since this mirror is primarily for large aperture application and also 

since the smallest F-number is 1/2, then z is a small number given by 


z < X/4f < V2D << 1 '13) 


For D = 20\, which is about the case computed before, z < 0.025. 

By introducing this approximation, the integration in (9) can be carried 

out in terms of basic functions and Fresnel integrals as shown in the 

following. Now 

2 3 4 

<P(z) = x-2 (f sin a +y) z + (f + 2 A x-2 cos a )z +2yz -2Axz 

5 6 7 

-2yz +2Axz +0(z). (14) 

Since P is in the neighborhood of F, Ax «f or x, and y « f . 

Further using the condition indicated by (13), one obtains the following 


(p(z) » x-2 (f sin a- + y) z + (f + 2Ax-2 cos a) z 

= A(z + B/2) 2 + (C-B 2 /4) = A U + D 
where A = f + 2Ax - 2 cos <*■; 

AB = -2(f sin a. + y) ; (15) 

AC = x 
U = z + B/2; 
D = C - B /4 
Similarly the other factor in the integrand of (10) can be approximated 

as follows, 

2 " 1/2 2 

(cos a ~ z S in a) (l + z ) -cos a - z sin a - z (cos a)/2 

+ z 3 (sin a)/2 = P(z) (16) 

Substituting (U-B/2) for z in P(z), one obtains a polynomial of third 

degree in U which may be written as 

2 3 
g(U) =a +bU+ c U+dU (17) 

when a, b, c, and d can be obtained by comparing (16) with (17). 



E l( p) = 2 Jg. e jk ° I 



g(U) e JkAU dU, 


U = -z + B/2 

U = z, + B/2 
2 1 


The first term of I is given by 



r 2 

J l = a J 
1 J 

e JkAU dU = 

J F( v/kA U ) - F( VkA U ) 


oo . 2 


where the Fresnel integral F(x) = e dx is available in tabulated form, 


The second tern of I is given by 


I = b 
2 J 

jkAU* jkAU* 

Ue JkAU dU = £ S Lf e 

2 jkA 

b sin kAz B jkA(z 2 + B 2 /4) 
______ e 


The third term of I is given by 

I, = C ^ U 2 e^ 1 ' 2 dB = ° 

3 , 2jkA 


^ UdeJ"" 2 



2 U 2 
[ (Ue JkAU ) 


r 2 2 

e JkAU dU 

°1 " u l 

j jkAU 

Ff /kAl I 
2 J 



The last term of I is given by 


. (" 2 TT 3 jkAU 2 dU = — — 

I = d J D e J 2jkA . 


U 2 d e JkAU 



U 2 e jkAU 

jkAU 2 
e J dU 


jkAU 2 jk AU 2 
(U 2 e - U x e 

± jkAU 2 jkAU^ 

) r-r (e - e 



To sum up the field at P near F is given by Equation (8) in which the 
field due to the first zone 

V P> - > /4 * ikD = ' p 

V P=l P 


z i> l 2> V X 4 are given b y (19) to (22) 

A, B, C } D, are given by (15) 

U U are given by (18b) and (10a); 

a, b, c, d are given by (16) and (17). 

In Equation (8), the focal length of the first zone is practically inde- 
pendent of frequency; however that of the rest zones for normal incidence can be 
easily found. In this case by symmetry the focus must lie on the axis. Let P 
be such a point on the axis with coordinates (Ax, o) . Then by definition 

2 2 2 
r = f + (Ax) + 2 f (Ax) cos 2t and 

Ax 1 Ax 2 A Ax 

r = f Q + =- cos 2T + - — 7t sin 2T + 9 — - 

o f 2 * \ f 


1" Vt 

Now the phase function of the field due to the n zone with n^l 
in Equation 8) becomes for a - 

CD = k S sin t - k r «. kx' - k f (1 + ^ cos 2*0 . (24) 
T n on n on 

However x' = (n-1) *. 
on o 

where p = an integer Then 

kx 7 

- k f (1 + . 




= P X 

<P = 2 TT (n-1) rfi— -27TP -^ (1 + -^£_ cos 2T) 

n a. /v j 

A. \r% Ax 

= - 2ir(p-n+l) — jr^_ - 2ir p — £— — ~" cos 2T * (25) 

Let ^ = ^ + d\ then 

<P = - 2 tt (p-n+1) (l-d\ X ) - 2 it p *■ /M cos 2t (Ax/f) 
n o 

= 2ir(p-n+l) + 2ir(p-n+l) d^\ -2irp C^o/M cos 2T (Ax/f) (26) 
Putting 2TT(p-n-^l) d\A ■ 2l7p (\yA) COS 2T (Ax/f), 
then Ax/f = [l - (n-1) / p] sec 2T d\^ 


Since x - (n-1) ^ = f(l-cos 2'«") , 
on o 

cos 2T _ i _ ( n _D X. /f = l - (n-1) / p, 


Thus the chromatic aberration expressed in fraction of the focal length is 

glVen by Ax/f = dV* (27) 


which is independent of n to the approximations assumed in (24). Therefore 

the best focal point for all the zones except the central one is at 

f(l + d^A ) d i a wavelength ^ 4- dV 
o o 


It is obvious that the relation (27) also holds for high mode operation 

when ^ = nA + d^ where m = integer. 


A coma-corrected cylindric mirror which has been studied previously for 
its coma aberration is re-examined for its chromatic aberration. Although 
such a device can be operated at multiple modes, at each mode it has a 
narrow frequency bandwidth. At a frequency 6.25% higher than the designed 
one, there is a slight loss in gain but the side lobes are raised to a 
very high level. At such a frequency the scanning performance becomes 
inferior to a corresponding smooth parabolic reflector. At a frequency 
12.5% higher than the designed one, the image formation becomes so poor 
that no focus is generally recognizable. It may be inferred that such a 
device has a total bandwidth of perhaps only five or six percent, depending 
on what deterioration can be tolerated on the side lobes. 

A simplified formula has been obtained for the image field in the 
neighborhood of the focus. Such a formula will be found particularly useful 
when the chromatic aberration is large. It may also be used to evaluate the 
field witn coma aberration at the design frequency if some errors at the minor 
lobes and nulls are tolerable. 

Although no computation has hitherto been done to show how close a 
solution this theory provides, yet it is clear that the major approximation 
involved is that only the geometric optics current is considered. Even in the 
absence of chromatic aberration, this has been previously shown to be a close 
approximat ion. 

A simple formula of the longitudinal chromatic aberration for the case 
of normal incidence is obtained. The lateral chromatic aberration for incident 
angle up to 20 and for a frequency range of + 6.257o is very small as indicated 
by the numerical results. 



1. S. Dasgupta and Y. T. Lo, "A Study of the Coma-corrected Zoned Mirror 
by Diffraction Theory." Technical Report No. 40, Ant. Lab., University 
of Illinois, July, 1959. 

For other references, see pp. 87 and 88 of the above report. 


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Cross Secti I'chni cal Report No. 29, J„ D. Dyson, 10 January 1958. 

"Non-Planar Logan I hrr.ically Periodic Antenna Structure," Tech nical Rep ort N^__3J 
D W Isbell. 20 February 1958 

"Electromagnetic Fields in Rectangular Slots," Technical Report No. 31 , N. J. 
Kuhn and P E Masl 10 March 1958 

The Efficiency of Excitation of a Surface Wave on a Dielectric Cylinder," 
Tech no . ■ Reporl J W. Duncan, 25 May 1958, 

"A Unidirectional Equiangular Spiral Antenna/' Technical Report No. 33, J. D. 
Dyson, 10 July 1958 

"Dielectric Coated Spheroidal Radiators," Technical Report No. 34 W. L Weeks 
12 September 1958. 

"A Theoretical Study of the Equiangular Spiral Antenna/' Technical Report 
No. 35, P E. Mast, 12 September 1958. ' ~~ 

Contract AF33 (616) -6079 

"Use of Coupled Waveguides in a Traveling Wave Scanning Antenna/' Technical 
Report No 36, R H. MacPhie, 30 April 1959. 

"On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and 
Infinite Ranges," Technical Report No. 37, Raj Mittra, 15 May 1959. 

"Prolate Spheroidal Wave Functions for Electromagnetic Theory/' Technical 
Report No. 38, W. L, Weeks, 5 June 1959. 

"Log Periodic Dipole Arrays," Technical Rep ort No. 39, D,E„ Isbell, 1 June 1959, 

"A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory " Technical 
Report No. 40, S. Dasgupta and Y. T. Lo, 17 July 1959, 

"The Radiation Pattern of a Dipole on a Finite Dielectric Sheet," Technical 
Report No, 41 , K, G, Balmain, 1 August 1959, 

"The Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem 
in a Waveguide," T echnical Report No. 42 , Raj Mittra, 1 October 1959. 

"impedance Properties of Complementary Mul titerminal Planar Structures," 
Technical Report No, 43, G. A, Deschamps, 11 November 1959. 

"On the Synthesis of Strip Sources," Technical Repor t No, 44, Raj Mittra, 
4 December 1959. 

"Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides,' 
Tech nical Report Jjo ^_45 , C. H. Tang and Y. T. Lo, 11 March 1960. 

"New Circularly Polarized Frequency Independent Antennas With Conical Beam or 
Omnidirectional Patterns,'' Technical Report No. 46, J.D c Dyson and P.E, Mayes, 
20 June 1960. 

"Logarithmically Periodic Resonant-V Arrays," Technical Report No. 47, P,E = 
Mayes and R, L, Carrel, 15 July 1960. 

~ Copies" available for a three week loan period. 
** Copies no longer available, 

AF 33(616) -6079 


One copy each unless otherwise indicated 


Iright Air Development Center 

Attn WCOSI, Library 
fright-Patterson Air Force Base, Ohio 


FS, Naval Air Test Center 

Attn: E7-315, Antenna Section 
Patuxent River, Maryland 


Bureau Naval Weapons 

Department of the Navy 

Attn (RR-13) 

Washington 25, D.C. 


Air Force Missile Test Center 

Attn, Technical Library 
Patrick Air Force Base, Florida 


Ballistics Research Lab. 

Attn Ballistics Measurement Lab. 

Aberdeen Proving Ground, Maryland 

"Hice of the Chief Signal Officer 
Attn.. SIGNET-5 

Eng , & Technical Division 
Washington 25. D,C, 

Rational Bureau of Standards 
department of Commerce 

Attn; Dr , A , G McNish 
Mungton 25, D.C 

• S. Navy Electronics Lab. 
! oi nt Loma 
n °iego 52, California 

ri ght Air Development Division 

Attn; NC Draganjac, WWDBEG 
r ight-Patterson AFB, Ohio (2 copies) 


USA White Sands Signal Agency 

White Sands Proving Command 

Attn: SIGWS-FC-02 
White Sands, New Mexico 


Air University Library 

Attn: AUL-8489 
Maxwell AFB, Alabama 

Army Rocket and Guided Missile Agency 
U.S. Army Ordnance Missile Agency 

Redstone Arsenal, Alabama 


Aero Space Technical Intelligence Center 

Attn: AFCIN-4c3b, Mr. Lee Roy Hay 
Wright-Patterson AFB, Ohio 


801st Air Division (SAC) 


DCTT Major Witry 

Lockbourne Air Force Base, Ohio 


Air University Library 

Attn: AUL-9642 
Maxwell Air Force Base, Alabama 


Bureau of Aeronautics 
Attn: Aer-EL-931 
Department of the Navy 
Washington 25, D.C. 

Armed Services Technical Information Agency 
Arlington Hall Station 

Arlington 12, Virginia (10 copies) 

(Excluding Top Secret and Restricted 
Data) (Reference AFR 205-43) 

AF 33(616)-6079 


Wright Air Development Center 
Attn: F. Behrens, WCLKR 
Wright-Patterson AFB, Ohio 


Air Research & Development Command 

Attn; RDTC 
Andrews Air Force Base 
Washington 25, D.C. 


Naval Research Laboratory 
Attn: Dr. A. Marston 
Washington 25, D.C. 

Director, Surveillance Dept . 
Evans Area 

Attn: Technical Document Center 
Belmar, New Jersey 


Hq , Air Force Cambridge Research Center 




Laurence G c Hans com Field 
Bedford, Massachusetts 


Air Proving Ground Command 

Attn. Classified Technical Data 
Branch D, 01 
Eglin Air Force Base, Florida 


Research and Development Command 

Hq . USAF 

Washington 25, D„C. 


Air Force Ballistics Missile Division 

Attn: Technical Library 
Air Force Unit Post Office 
Los Angeles 45, California 


Air Force Missile Development Center 

Attn Technical Library 
Holloman Air Force Base, New Mexico 


801st Air Division (SAC) 

Attn DCTTD, Major Hougan 
Lockbourne Air Force Base, Ohio 


Rome Air Development Center 


RCERA-1, M, Diab 

Griffiss Air Force Base, New York 

Chief, Bureau of Ships 
Department of the Navy 

Attn: Code 838D 
Washington 25, D.C. 

Commanding Officer & Director 
U.S. Navy Electronics Laboratory 

Attn: Library 
San Diego 52, California 

Andrew Alford Consulting Engineers 
Attn: Dr, A. Alford 

M./F Contract AF33(600)-36108 
299 Atlantic Avenue 
Boston 10, Massachusetts 

ATA Corporation 
1200 Duke Street 
Alexandria, Virginia 

Bell Telephone Labs., Inc. 
Attn: R. L. Mattingly 

M/F Contract AF33(616)-5499 
Whippany, New Jersey 

Bendix Radio Division 
Bendix Aviation Corporation 
Attn: Dr. K. F. Umpleby 

M/F Contract AF33(600)-35407 
Towson 4, Maryland 

Boeing Airplane Company 
Attn: C. Armstrong 

M/F Contract AF33(600)-36319 
7755 Marginal Way 
Seattle, Washington 

Boeing Airplane Company 
Attn: Robert Shannon 

M, F Contract AF33( 600)-35992 
Wichita. Kansas 

F 33(616)-6079 

anoga Corporation 

! /F Contract AF08( 603)-4327 

955 Sepulveda Boulevard 

.0. Box 550 

an Nuys, California 

Dome & Margolin, Inc. 

M/F Contract AF33(600)-35992 

30 Sylvester Street 


Long Island, New York 

r. C, H Papas 

epartment of Electrical Engineering 
alifornia Institute of Technology 
asadena, California 

hance-Vought Aircraft Division 

nited Aircraft Corporation 
Attn- R C , Blaylock 
THRL' BuAer Representative 

M F Contract NOa(s) 57-187 

alias, Texas 

ollins Radio Company 
Attn Dr , R, H, DuHamel 

M F Contract AF33( 600) -37559 
edar Rapids, Iowa 

Attn. R . Honer 

M F Contract AF33(600)-26530 
an Diego Division 
an Diego 12, California 

Douglas Aircraft Co., Inc. 
Attn: G. ? Rilley 

M/F Contract AF33(600)-25669 & 
Tulsa, Oklahoma 

Exchange and Gift Division 
The Library of Congress 

Washington 25, D.C . 

(2 copies) 

Fairchild Engine & Airplane Corp. 
Fairchild Aircraft Division 
Attn: Engineering Library 
S. Rolfe Gregory 
M/F Contract AF33(038)-18499 
Hagerstown, Maryland 

Dr. Frank Fu Fang 

Boeing Airplane Company 

Transport Division, Physical Research 

Renton, Washington 

General Electric Company 

Attn D, H. Kuhn, Electronics Lab, 

M/F Contract AF30(635)-12720 
Building 3, Room 301 


ort Worth Division 
Attn, C, R Curnutt 

M F Contract AF33(600)-32841 & College Park 

AF33(600)-31625 113 S, Salina Street 
ort Worth, Texas Syracuse, New York 

Apartment of Electrical Engineering 

Attn Dr . H G Booker 

ornell University 
thaca, New York 

'niversity of Denver 
>enver Research Institute 
diversity Park 
'enver 10. Colorado 

lalmo Victor Company 

Attn Engineering Technical Library 

M F Contract AF33(600)-27570 
515 Industrial Wav 
ielmont, California 

General Electronic Laboratories, Inc. 
Attn; F„ Parisi 

M/F Contract AF33(600)-35796 
18 Ames Street 
Cambridge 42, Massachusetts 

Goodyear Aircraft Corporation 
Attn: G, Welch 

M/F Contract AF33(616)-5017 
Akron, Ohio 

Granger Associates 
M/F Contract AF19(604)-5 509 
966 Commercial Street 
Palo Alto, California 

AF 33(616)-6079 

Grumman Aircraft Engineering Corp. 
Attn: J, S, Erickson 

Asst, Chief, Avionics Dept . 

M. F Contract NOa(s) 51-118 
Bethpage, Long Island, New York 

ITT Laboratories 

Attn; A. Kandoian 

M/F Contract AF33(616)-5120 
500 Washington Avenue 
Nutley 10, New Jersey 

Gulton Industries, Inc. 
Attn; B Bittner 

M F Contract AF33( 600)-36869 
P„0 o Box 8345 
15000 Central, East 
Albuquerque, New Mexico 

Hallicraf ters Corporation 
Attn„ D, Herling 

M F Contract AF33(604)-21260 
440 W, Fifth Avenue 
Chicago, Illinois 

Technical Reports Collection 
Attn; Mrs E. L, Hufschmidt 
303 A, Pierce Hall 
Harvard University 
Cambridge 38. Massachusetts 

Hoffman Laboratories, Inc. 

Attn; S, Varian (for Classified) 
Technical Library (for 

M F Contract AF33(604)-17231 
Los Angeles, California 

Dr. R, F Hyneman 

P.O, Box 2097 

Mail Station C-152 

Building 600 

Hughes Ground Systems Group 

Fullerton, California 

ITT Laboratories 
Attn: L. DeRosa 

M/F Contract AF33(616)-5120 
500 Washington Avenue 
Nutley 10, New Jersey 

ITT Laboratories 

A Div. of Int. Tel. & Tel. Corp, 
Attn: G, S. Giffin, ECM Lab. 
3700 E. Pontiac Street 
Fort Wayne, Indiana 

Jansky and Bailey, Inc. 
Engineering Building 

Attn; Mr. D. C. Ports 
1339 Wisconsin Avenue, N.W. 
Washington, D.C. 

Jasik Laboratories, Inc. 
100 Shames Drive 
Westbury, New York 

John Hopkins University 
Radiation Laboratory 
Attn; Librarian 

M/F Contract AF33(616)-68 
1315 St. Paul Street 
Baltimore 2, Maryland 

Applied Physics Laboratory 
Johns Hopkins University 
8621 Georgia Avenue 
Silver Spring, Maryland 

HRB-Singer, I nc 

Attn Mr R A Evans 
Science Park 
State College, Pa , 

Mr, Dwight Isbell 
4620 Sunnyside 
Seattle 3, Washington 

Lincoln Laboratories 
Attn. Document Room 

M/F Contract AF19(122)-458 
Massachusetts Institute of Technology 
P.O. Box 73 
Lexington 73, Massachusetts 

AF 33(616)-6079 

Litton Industries, Inc. 
Maryland Division 

Attn,. Engineering Library 

M F Contract AF33(600)-37292 
4900 Calvert Road 
College Park, Maryland 

Lockheed Aircraft Corporation 
Attn; C. D, Johnson 

M F Contract NOa(s) 55-172 
P.O, Box 55 
Burbank, California 

University of Michigan 
Aeronautical Research Center 
Attn: Dr, K. Seigel 

M/F Contract AF30( 602)-1853 
Willow Run Airport 
Ypsilanti, Michigan 

Microwave Radiation Co., Inc. 
Attn: Dr. M, J. Ehrlich 

M/F Contract AF33(616)-6528 
19223 S. Hamilton Street 
Gardena, California 

Lockheed Missiles & Space Division 
Attn E . A . Blasi 

M F Contract AF33(600)-28692 & 
Department 58-15 
Plant 1, Building 130 
Sunnyvale, California 

The Martin Company 

Attn; W, A Kee, Chief Librarian 

M. F Contract AF33(600)-37705 
Library & Document Section 
Baltimore 3, Maryland 

Ennis Kuhlman 

McDonnell Aircraft 

P.O. Box 516 

Lambert Municipal Airport 

St. Louis 21, Missouri 

Mel par, Inc , 

Attn: Technical Library 

M, F Contract AF19(604)-4988 
Antenna Laboratory 
3000 Arlington Blvd 
Falls Church. Virginia 

Melville Laboratories 
Walt Whitman Road 
Melville, Long Island, 
New York 

Motorola, Inc. 

Attn: R, C. Huntington 
8201 E. McDowell Road 
Phoenix, Arizona 

Physical Science Lab. 

Attn. R. Dressel 
New Mexico College of A and MA 
State College, New Mexico 

North American Aviation, Inc 

Attn: J. D„ Leonard, Eng . Dept . 
M/F Contract NOa(s) 54-323 
4300 E. Fifth Avenue 
Columbus, Ohio 

North American Aviation, Inc. 
Attn: H, A. Storms 

M/F Contract AF33( 600)- 36599 
Department 56 
International Airport 
Los Angeles 45, California 

Northrop Aircraft, Inc. 

Attn: Northrop Library, Dept. 2135 
M/F Contract AF33 (600) -27679 
Hawthorne, California 

Dr . R . E . Beam 
Microwave Laboratory 
Northwestern University 
Evanston, Illinois 

AF 33(616) -6079 

Ohio State University Research 

Attn: Dr. T, C„ Tice 

M F Contract AF33(616)-6211 
1314 Kinnear Road 
Columbus 8, Ohio 

University of Oklahoma Res. Inst. 
Attn: Prof, C, L. Farrar 

M/F Contract AF33(616)-5490 
Norman, Oklahoma 

Dr. D. E. Royal 

Ramo-Wooldridge, a division of Thompson 

Ramo Wooldridge Inc. 

8433 Fallbrook Avenue 

Canoga Park, California 

Rand Corporation 
Attn: Librarian 

M/F Contract AF18(600)-1600 
1700 Main Street 
Santa Monica. California 

Philco Corporation 

Government and Industrial Division 
Attn: Dr„ Koehler 

M/F Contract AF33(616)-5325 
4700 Wissachickon Avenue 
Philadelphia 44, Pennsylvania 

Prof. A, A. Oliner 
Microwave Research Institute 
Polytechnic Institute of Brooklyn 
55 Johnson Street - Third Floor 
Brooklyn, New York 

Radiation, Inc , 
Technical Library Section 
Attn Antenna Department 

M. F Contract AF33(600)-36705 
Melbourne, Florida 

Radio Corporation cf America 
RCA Laboratories Division 
Attn: Librarian 

M/F Contract AF33(616)-3920 
Princeton, New elersey 

Radioplane Company 

M/F Contract AF33( 600)-23893 

Van Nuys, California 

Ramo-Wooldridge, a division of 
Thompson Ramo Wooldridge, Inc. 

Attn. Technical Information Services 
8433 Fallbrook Avenue 
P.O. Box 1006 
Canoga Park, California 

Rantec Corporation 

Attn: R. Krausz 
M/F Contract AF19(604)-3467 
Calabasas, California 

Raytheon Manufacturing Corp. 
Attn: Dr. R. Borts 

M/F Contract AF33(604)-15634 
Wayland, Massachusetts 

Republic Aviation Corporation 
Attn: Engineering Library 

M/F Contract AF33(600)-34752 
Long Island, New York 

Republic Aviation Corporation 
Guided Missiles Division 
Attn: J. Shea 

M/F Contract AF33(616)-5925 
223 Jericho Turnpike 
Mineola, Long Island, New York 

Sanders Associates, Inc. 
95 Canal Street 

Attn: Technical Library 
Nashua, New Hampshire 

Smyth Research Associates 

Attn: J. B. Smyth 
3555 Aero Court 
San Diego 11, California 

Space Technology Labs, Inc. 

Attn: Dr. R. C. Hansen 
P.O. Box 95001 
Los Angeles 45, California 
M/F Contract AF04( 647)-361 

\F 33(616^-6079 

Sperry Gyroscope Company 
Attn B, Berkowitz 

M F Contract AF33( 600)-28107 
areat Neck 
Long Island, New '-'ork 

Stanford Electronics Laboratory 
Attn; Applied Electronics Lab. 
Document Library 
Stanford University 
Stanford, California 

Stanford Research Institute 

Attn: Mary Lou Fields, Acquisitions 
Documents Center 
Menlo Park. California 

Stanford Research T nstitute 
Aircraft Radiation Systems Lab. 
Attn D, Scheuch 

M F Contract AF33(616)-5584 
Menlo Park, California 

Sylvania Electric Products, Inc. 
Electronic Defense Laboratory 
M/F Contract DA 36-039-SC-75012 
P.O. Box 205 
Mountain View, Talifornia 

Mr. Roger Battie 

Supervisor, technical Liaison 

Sylvania Electric Products, Inc. 

Electronic Systems Division 

P.O. Box 188 " 

Mountain View, California 

Sylvania Electric Products, Inc. 
Electric Svs terns Division 
Attn P Faflick 

M F Contract AF33(038)-21250 
100 First S+reet 
Waltham 54. Massachusetts 

Tamar Electronics, Inc 
Attn L B McMurren 
2045 w Rosecrans Avenue 
Gardena, California 

Technical Research Group 
M/F Contract AF33(616)-6093 
2 Aerial Way 
Syosset, New York 

Temco Aircraft Corporation 
Attn: G. Cramer 

M/F Contract AF33(600)-36145 
Garland, Texas 

Electrical Engineering Res. Lab, 
University of Texas 
Box 8026, University Station 
Austin, Texas 

A. S, Thomas, Inc, 
M/F Contract AF04(645)-30 
161 Devonshire Street 
Boston 10, Massachusetts 

Westinghouse Electric Corporation 
Air Arm Division 
Attn: P. D, Newhouser 

Development Engineering 
M/F Contract AF33( 600)-27852 
Friendship Airport 
Baltimore, Maryland 

Professor Morris Kline 

Institute of Mathematical Sciences 

New York University 

25 Waverly Place 

New York 3, New York 

Dr , S. Dasgupta 

Government Engineering College 

Jabalpur, M.P, 


Dr.. Richard C. Becker 
10829 Berkshire 
Westchester, Illinois 

The Engineering Library 
Princeton University 
Princeton, New Jersey 

AF 33(616)-6079 

Dr. B. Chatterjee 
Communication Engineering Dept . 
Indian Institute of Technology 
Kharagpur (S.E. Rly.) 

Sperry Phoenix Company 
Attn: Technical Librarian 
P.O. Box 2529 
21111 North 19th Avenue 
Phoenix, Arizonia 

Dr. Harry Letaw, Jr. 

Raytheon Company 

Surface Radar and Navigation Operations 

State Road West 

Wayland, Massachusetts